Final answer:
a. The cardinality of the set ≤ 10 is 21. b. The cardinality of the set {x ∈ Z: 1≤x² ≤ 2} is 2. c. Please provide additional clarification on ∅3. d. The cardinality of the set {x ∈ Z: ∅ ∈ x} is 0. e. More information is needed to determine which probability will be higher. f. Detailed calculations are needed to find probabilities. g. The distribution for X is not exponential. h. The cardinality of the set {{1, 2}, {3, 4, 5}} is 2.
Step-by-step explanation:
a. The set x ∈ Z: consists of all integers that are less than or equal to 10 in absolute value. This set can be written as {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Therefore, the cardinality of this set is 21.
b. The set {x ∈ Z: 1 ≤ x² ≤ 2} consists of all integers whose square falls between 1 and 2, inclusive. The square root of 2 is approximately 1.414, so the integers that satisfy this condition are 1 and -1. Therefore, the cardinality of this set is 2.
c. The notation ∅3 is not clear. Please provide additional clarification.
d. The notation ∅ (empty set) is a set that does not contain any elements. Therefore, the set {x ∈ Z: ∅ ∈ x} would also be an empty set with a cardinality of 0.
e. It is not possible to determine which probability will be higher without additional information or calculation.
f. More information is needed to calculate probabilities.
g. The distribution for X is not exponential because an exponential distribution follows a specific probability density function, which is not evident in the provided information.
h. The set {{1, 2}, {3, 4, 5}} consists of two subsets: {1, 2} and {3, 4, 5}. Therefore, the cardinality of this set is 2.